Additive limits

In graph theory, the recently developed theory of graph limits has proven to be a useful tool for analysing large dense graphs, being a convenient reformulation of the Szemerédi regularity lemma. Roughly speaking, the theory asserts that given any sequence of finite graphs, one can extract a subsequence which converges (in a specific sense) to a continuous object known as a “graphon” – a symmetric measurable function . What “converges” means in this context is that subgraph densities converge to the associated integrals of the graphon . For instance, the edge density

converge to the integral

the triangle density

converges to the integral

the four-cycle density

converges to the integral

and so forth. One can use graph limits to prove many results in graph theory that were traditionally proven using the regularity lemma, such as the triangle removal lemma, and can also reduce many asymptotic graph theory problems to continuous problems involving multilinear integrals (although the latter problems are not necessarily easy to solve!). See this text of Lovasz for a detailed study of graph limits and their applications.

One can also express graph limits (and more generally hypergraph limits) in the language of nonstandard analysis (or of ultraproducts); see for instance this paper of Elek and Szegedy, Section 6 of this previous blog post, or this paper of Towsner. (In this post we assume some familiarity with nonstandard analysis, as reviewed for instance in the previous blog post.) Here, one starts as before with a sequence of finite graphs, and then takes an ultraproduct (with respect to some arbitrarily chosen non-principal ultrafilter ) to obtain a nonstandard graph , where is the ultraproduct of the , and similarly for the . The set can then be viewed as a symmetric subset of which is measurable with respect to the Loeb -algebra of the product (see this previous blog post for the construction of Loeb measure). A crucial point is that this -algebra is larger than the product of the Loeb -algebra of the individual vertex set . This leads to a decomposition

where the “graphon” is the orthogonal projection of onto , and the “regular error” is orthogonal to all product sets for . The graphon then captures the statistics of the nonstandard graph , in exact analogy with the more traditional graph limits: for instance, the edge density

(or equivalently, the limit of the along the ultrafilter ) is equal to the integral

where denotes Loeb measure on a nonstandard finite set ; the triangle density

(or equivalently, the limit along of the triangle densities of ) is equal to the integral

and so forth. Note that with this construction, the graphon is living on the Cartesian square of an abstract probability space , which is likely to be inseparable; but it is possible to cut down the Loeb -algebra on to minimal countable -algebra for which remains measurable (up to null sets), and then one can identify with , bringing this construction of a graphon in line with the traditional notion of a graphon. (See Remark 5 of this previous blog post for more discussion of this point.)

Additive combinatorics, which studies things like the additive structure of finite subsets of an abelian group , has many analogies and connections with asymptotic graph theory; in particular, there is the arithmetic regularity lemma of Green which is analogous to the graph regularity lemma of Szemerédi. (There is also a higher order arithmetic regularity lemma analogous to hypergraph regularity lemmas, but this is not the focus of the discussion here.) Given this, it is natural to suspect that there is a theory of “additive limits” for large additive sets of bounded doubling, analogous to the theory of graph limits for large dense graphs. The purpose of this post is to record a candidate for such an additive limit. This limit can be used as a substitute for the arithmetic regularity lemma in certain results in additive combinatorics, at least if one is willing to settle for qualitative results rather than quantitative ones; I give a few examples of this below the fold.

It seems that to allow for the most flexible and powerful manifestation of this theory, it is convenient to use the nonstandard formulation (among other things, it allows for full use of the transfer principle, whereas a more traditional limit formulation would only allow for a transfer of those quantities continuous with respect to the notion of convergence). Here, the analogue of a nonstandard graph is an ultra approximate group in a nonstandard group , defined as the ultraproduct of finite -approximate groups for some standard . (A -approximate group is a symmetric set containing the origin such that can be covered by or fewer translates of .) We then let be the external subgroup of generated by ; equivalently, is the union of over all standard . This space has a Loeb measure , defined by setting

whenever is an internal subset of for any standard , and extended to a countably additive measure; the arguments in Section 6 of this previous blog post can be easily modified to give a construction of this measure.

The Loeb measure is a translation invariant measure on , normalised so that has Loeb measure one. As such, one should think of as being analogous to a locally compact abelian group equipped with a Haar measure. It should be noted though that is not actually a locally compact group with Haar measure, for two reasons:

There is not an obvious topology on that makes it simultaneously locally compact, Hausdorff, and -compact. (One can get one or two out of three without difficulty, though.)

The addition operation is not measurable from the product Loeb algebra to . Instead, it is measurable from the coarser Loeb algebra to (compare with the analogous situation for nonstandard graphs).

Nevertheless, the analogy is a useful guide for the arguments that follow.

Let denote the space of bounded Loeb measurable functions (modulo almost everywhere equivalence) that are supported on for some standard ; this is a complex algebra with respect to pointwise multiplication. There is also a convolution operation , defined by setting

whenever , are bounded nonstandard functions (extended by zero to all of ), and then extending to arbitrary elements of by density. Equivalently, is the pushforward of the -measurable function under the map .

The basic structural theorem is then as follows.

Theorem 1 (Kronecker factor) Let be an ultra approximate group. Then there exists a (standard) locally compact abelian group of the form

for some standard and some compact abelian group , equipped with a Haar measure and a measurable homomorphism (using the Loeb -algebra on and the Baire-algebra on ), with the following properties:

(i) has dense image, and is the pushforward of Loeb measure by .

(ii) There exists sets with open and compact, such that

(iii) Whenever with compact and open, there exists a nonstandard finite set such that

(iv) If , then we have the convolution formula

where are the pushforwards of to , the convolution on the right-hand side is convolution using , and is the pullback map from to . In particular, if , then for all .

One can view the locally compact abelian group as a “model “or “Kronecker factor” for the ultra approximate group (in close analogy with the Kronecker factor from ergodic theory). In the case that is a genuine nonstandard finite group rather than an ultra approximate group, the non-compact components of the Kronecker group are trivial, and this theorem was implicitly established by Szegedy. The compact group is quite large, and in particular is likely to be inseparable; but as with the case of graphons, when one is only studying at most countably many functions , one can cut down the size of this group to be separable (or equivalently, second countable or metrisable) if desired, so one often works with a “reduced Kronecker factor” which is a quotient of the full Kronecker factor . Once one is in the separable case, the Baire sigma algebra is identical with the more familiar Borel sigma algebra.

Given any sequence of uniformly bounded functions for some fixed , we can view the function defined by

as an “additive limit” of the , in much the same way that graphons are limits of the indicator functions . The additive limits capture some of the statistics of the , for instance the normalised means

converge (along the ultrafilter ) to the mean

and for three sequences of functions, the normalised correlation

converges along to the correlation

the normalised Gowers norm

converges along to the Gowers norm

and so forth. We caution however that some correlations that involve evaluating more than one function at the same point will not necessarily be preserved in the additive limit; for instance the normalised norm

does not necessarily converge to the norm

but can converge instead to a larger quantity, due to the presence of the orthogonal projection in the definition (4) of .

An important special case of an additive limit occurs when the functions involved are indicator functions of some subsets of . The additive limit does not necessarily remain an indicator function, but instead takes values in (much as a graphon takes values in even though the original indicators take values in ). The convolution is then the ultralimit of the normalised convolutions ; in particular, the measure of the support of provides a lower bound on the limiting normalised cardinality of a sumset. In many situations this lower bound is an equality, but this is not necessarily the case, because the sumset could contain a large number of elements which have very few () representations as the sum of two elements of , and in the limit these portions of the sumset fall outside of the support of . (One can think of the support of as describing the “essential” sumset of , discarding those elements that have only very few representations.) Similarly for higher convolutions of . Thus one can use additive limits to partially control the growth of iterated sumsets of subsets of approximate groups , in the regime where stays bounded and goes to infinity.

Theorem 1 can be proven by Fourier-analytic means (combined with Freiman’s theorem from additive combinatorics), and we will do so below the fold. For now, we give some illustrative examples of additive limits.

Example 2 (Bohr sets) We take to be the intervals , where is a sequence going to infinity; these are -approximate groups for all . Let be an irrational real number, let be an interval in , and for each natural number let be the Bohr set

In this case, the (reduced) Kronecker factor can be taken to be the infinite cylinder with the usual Lebesgue measure . The additive limits of and end up being and , where is the finite cylinder

and is the rectangle

Geometrically, one should think of and as being wrapped around the cylinder via the homomorphism , and then one sees that is converging in some normalised weak sense to , and similarly for and . In particular, the additive limit predicts the growth rate of the iterated sumsets to be quadratic in until becomes comparable to , at which point the growth transitions to linear growth, in the regime where is bounded and is large.

If were rational instead of irrational, then one would need to replace by the finite subgroup here.

Example 3 (Structured subsets of progressions) We take be the rank two progression

where is a sequence going to infinity; these are -approximate groups for all . Let be the subset

Then the (reduced) Kronecker factor can be taken to be with Lebesgue measure , and the additive limits of the and are then and , where is the square

and is the circle

Geometrically, the picture is similar to the Bohr set one, except now one uses a Freiman homomorphism for to embed the original sets into the plane . In particular, one now expects the growth rate of the iterated sumsets and to be quadratic in , in the regime where is bounded and is large.

Example 4 (Dissociated sets) Let be a fixed natural number, and take

where are randomly chosen elements of a large cyclic group , where is a sequence of primes going to infinity. These are -approximate groups. The (reduced) Kronecker factor can (almost surely) then be taken to be with counting measure, and the additive limit of is , where and is the standard basis of . In particular, the growth rates of should grow approximately like for bounded and large.

Example 5 (Random subsets of groups) Let be a sequence of finite additive groups whose order is going to infinity. Let be a random subset of of some fixed density . Then (almost surely) the Kronecker factor here can be reduced all the way to the trivial group , and the additive limit of the is the constant function . The convolutions then converge in the ultralimit (modulo almost everywhere equivalence) to the pullback of ; this reflects the fact that of the elements of can be represented as the sum of two elements of in ways. In particular, occupies a proportion of .

Example 6 (Trigonometric series) Take for a sequence of primes going to infinity, and for each let be an infinite sequence of frequencies chosen uniformly and independently from . Let denote the random trigonometric series

Then (almost surely) we can take the reduced Kronecker factor to be the infinite torus (with the Haar probability measure ), and the additive limit of the then becomes the function defined by the formula

In fact, the pullback is the ultralimit of the . As such, for any standard exponent , the normalised norm

can be seen to converge to the limit

The reader is invited to consider combinations of the above examples, e.g. random subsets of Bohr sets, to get a sense of the general case of Theorem 1.

for some standard ; we abbreviate the latter inclusion as . By an ultra coset progression, we mean the sumset of a nonstandard finite group and a nonstandard generalised arithmetic progression

with (known as the rank) standard, the generators in and the dimensions being nonstandard natural numbers. (To get the containment , one can first use the Bogulybov lemma to get a large ultra coset progression inside , so that can be covered by translates of ; one can then add these translates to the generators of to obtain an ultra coset progression with the required properties.

We call the ultra coset progression -proper if the sums for and for are all distinct. If fails to be -proper, then we can find a containment

where the coset progression has strictly smaller rank than ; see e.g. Lemma 5.1 of this paper of Van Vu and myself). Iterating this fact, we see that we may assume without loss of generality that is -proper. In particular, the group can now be parameterised by the sums with for , with each element of having exactly one representation of this form.

The dimensions are either bounded (and thus standard natural numbers) or unbounded. After permuting the generators if necessary, we may assume that are unbounded and are bounded for some with . We then have an external surjective group homomorphism defined by

this will end up being the non-compact portion of the projection map that we will eventually construct. The image is precompact in (in fact it is compact, thanks to countable saturation).

Now we perform some Fourier analysis on (analogous to the usual theory of Fourier analysis on locally compact abelian groups). Define a frequency to be a measurable homomorphism from to , and let denote the space of such frequencies; this is an additive group, which should be thought of as a “Pontryagin dual” to (even though is not a locally compact group). Meanwhile, we have the (genuine) Pontryagin dual of , using the identification

The homomorphism then induces a dual homomorphism , defined by the formula

for all and . This homomorphism is easily seen to be injective. If we let denote the cokernel of this map, then is an abelian group (which we will view as a discrete group) and we have the short exact sequence

Observe that is a divisible group. From this and a Zorn’s lemma argument we can split this short exact sequence, lifting up to a subgroup of , so that the latter group can be viewed as the direct sum of and .

Let be the Pontryagin dual of , that is to say the space of all homomorphisms from to (with no measurability or continuity hypotheses imposed). This is a compact abelian group (it is a closed subset of , which is compact by Tychonoff’s theorem). Set . We have a homomorphism , defined by

We claim that has dense image. Since is surjective, it suffices to show that the map from to has dense image from to , where

is the kernel of . The closure of the image of is a compact subgroup of , so this map did not have dense image, there would exist a non-trivial in the Pontryagin dual of which annihilates all of . The map then factors through and thus can be identified with an element of ; but and only intersect at , a contradiction. Thus has dense image.

It is a routine matter to verify that is measurable (noting that the Baire algebra is generated by the product of measurable sets in and cylinder sets in ), that is precompact, and that the inverse image of any compact set is contained in for some standard . From this and the Riesz representation theorem, we can define a Haar measure on by defining

for all continuous, compactly supported functions ; the translation invariance of this measure follows from the surjectivity of . From Urysohn’s lemma and the inner and outer regularity of Haar measures, one can then show that is the pushforward of Loeb measure under .

Now we show the convolution property (3). First suppose that , which in particular implies that

for all , since the function factors through . By the Loeb measure construction, we can write as the limit (in ) of functions , where are uniformly bounded nonstandard functions and is some standard natural number. Then we have

which in particular implies that

where ranges over all nonstandard maps of the form

for some and nonstandard homomorphism . From (nonstandard) Fourier analysis, we conclude that

for any bounded nonstandard function , or equivalently that

and thus on taking limits we see that , and on taking further limits we see that for any , as required. This proves (3) when ; similarly when . To finish off the general case of (3), it suffices to show that

for bounded measurable . By Fourier decomposition, we may assume that takes the form

for some and some continuous compactly supported , and similarly

for some and continuous compactly supported .

If , then for some , and one can use this to show that and both vanish. Thus we may assume that ; using modulation symmetry we may then assume that . It thus suffices to show that

A direct calculation shows that the left and right hand sides agree up to constants; but both sides also have integral when integrated against , so they must agree identically.

Now, we prove the inclusions (1). The outer inclusion comes from the compactness (or precompactness) of . For the inner inclusion, we note from (3) and the positive measure and symmetry of that is the pullback of a continuous function on that is positive at the origin, and thus also bounded away from zero on a neighbourhood of the origin. This implies that the set has full measure in . We then let be a smaller symmetric neighbourhood of the origin such that . We then see that for any , the sets and both have full measure in , and hence lies in . This gives the inner inclusion of (1).

Finally, we show the regularity claim (2). Given , we may apply Urysohn’s lemma to find non-negative bounded continuous functions such that is supported in and is at least on . Letting be the pullbacks of by , we conclude using (3) that is at least on and vanishes outside of . Approximating in by bounded nonstandard functions supported in , we conclude that is at least (say) on and less than (say) outside of . If one then sets to be the non-standard set where , we obtain the claim.

— 2. Sample applications of theorem —

In this section we illustrate how this theorem can be used to reprove some existing results in additive combinatorics, reducing them to various statements in continuous harmonic analysis. We begin with a qualitative version of a result of Croot and Sisask on almost periods, which reduces to the classical fact that the convolution of two square-integrable functions is continuous.

Proposition 7 (Croot-Sisask) Let be a -approximate group in an additive group , let be subsets of , and let . Then there exists a subset of with such that

for all (using the non-normalised convolution ).

The Croot-Sisask argument in fact gives a quantitative lower bound of exponential type on , but such bounds are not available through the qualitative limiting arguments given here. The Croot-Sisask argument also works in non-commutative groups ; it is likely the arguments here would also extend to that setting once one developed a non-commutative version of Theorem 1, but we have not investigated that here.

Proof: By the usual transfer arguments, it suffices to show that when is an ultra approximate group, are non-standard subsets of , and , there exists a nonstandard subset of with such that

for all (using the normalised convolution ). But by (3), is the pullback via of a continuous compactly supported function, so (5) holds for in for some neighbourhood of the identity, and thus by (2) it also holds for all in some nonstandard of positive Loeb measure. The claim follows.

Now we give a proof of Roth’s theorem (in the averaged form of Varnavides), at least for groups of odd order.

Proposition 8 (Roth’s theorem) Let , let be a finite group of odd order, and let be such that . Then there are pairs such that .

Proof: By the usual transfer arguments, it suffices to show that when is a nonstandard finite group of odd order, and is a nonstandard subset of with , then there are pairs such that ; equivalently, we need to show that

where . As has positive measure, is not identically zero. By a version of the Lebesgue differentiation theorem, we can then find a point in the Kronecker factor group such that has positive density on every precompact neighbourhood of , and is bounded away from zero on a subset of a symmetric open precompact neighbourhood of of density greater than . From this and (3) we see that is bounded away from zero on almost all of for some neighbourhood of . Also, as has odd order, the map is a measure-preserving map on , it must also be so on , and so we conclude that has positive measure in , and (6) follows.

Proposition 9 For every there is such that if is such that , then there is an arithmetic progression such that and .

Proof: Here, we perform the transfer step more explicitly, as it is slightly trickier here. Suppose for contradiction that the claim failed, then there exists an and a sequence with

but such that there is no arithmetic progression with such that . Note that must go to infinity (otherwise one could take to consist of a single element of , which must be non-empty from (7). Taking ultraproducts, we arrive at a nonstandard subset of for some unbounded natural number , such that

and such that there is no nonstandard arithmetic progression with and (say). Here is the Loeb measure associated with the approximate group and the group that it generates. By inspection of the proof of Theorem 1, the Kronecker factor of this group can be taken to be for some compact group , with projection given by for some measurable map , and Haar measure given by the product of Lebesgue measure on and the Haar probability measure on . If we let , then is supported in and takes values in , and from (3) and (8) we see that the set

is such that

We can view as a measurable function , defined by , and similarly the indicator function can be viewed as a measurable function defined by . Being measurable, may be approximated in by piecewise constant functions. One can then adapt the proof of the Lebesgue differentiation theorem to show that almost all are a Lebesgue point of , in the sense that

Similarly, almost all is a Lebesgue point of in the sense that

From this, we see that for almost all , we have the inclusion

up to null sets in , where the convolution is now with respect to the Haar probability measure . On the other hand, from Fubini’s theorem we have

and

Also is supported on . Thus by the pigeonhole principle, we may find an such that

and such that (9) holds up to null sets, and such that is a Lebesgue point for . If we fix this and now set and , we thus have

At this point it is convenient to split the compact abelian group as

where is the connected component of the identity, thus is a totally disconnected group. Let be the pushforward of to via the projection map , thus is a measurable function of total integral . We claim that

To see this, suppose for contradiction that

We may disintegrate

where is a measurable map from to finite measures on , such that for almost every , is supported on and has total mass . For almost every and , we then have that

is supported in . By Fubini’s theorem we have

where is the Haar probability measure on . From (10), we conclude that for almost every , there is a positive measure set of such that

for a positive measure set of (in particular ), and that is supported in . On the other hand, and are supported on sets of measure at least and . Applying Kemperman’s theorem (see this previous post) , we conclude that

for almost every with , and for a positive measure set of . But this leads to a contradiction if we take to be within of the essential supremum of . This proves (11).

As is totally disconnected, we can express the origin as the intersection of open subgroups. From this, (11), and a Lebesgue differentiation argument, we may find a coset of an open subgroup of such that

Letting be a pullback of to , we thus have

Since is a Lebesgue point for , we may thus find a neighbourhood of in such that

or equivalently

To finish the proof of the claim, it then suffices to show that differs from a nonstandard arithmetic progression by a set of arbitrarily small Loeb measure.

Consider the quotient homomorphism formed by first using to project to , then projecting to , then to . This is a Loeb measurable map, and thus the pointwise limit (up to null sets) by a nonstandard function. But observe that for , one has if and only if for almost every . In particuar, if is a nonstandard function which is sufficiently close to , then if and only if is the most common value of for . Using this, one can find a representative of that is precisely a nonstandard function on (say). Thus is now a nonstandard map from to the standard finite group , and from construction one can check that for all (and not merely almost all) . From this it is easy to see that is periodic with some bounded period, and that the level sets of are infinite nonstandard arithmetic progressions of bounded spacing. The claim then follows.

13 comments

* “What “converges” means in this context means that” –> “What “converges” means in this context is that”
* Use “” and “”, reps., for absolut values instead of “” to get correct horixontal spacing
* Should there be four integrals instead of three in the sixth displayed expression?
* Use “” instead of “” to get correct horixontal spacing

Consider a hyperfinite graph G for which some rescaling of the graph distance of G has the property that its infinitesimal collapse is locally compact. (infinitesimal collapse means the space obtained identifying elements infinitely close to each other — which yields always a complete metric space in which of course distances between points may be infinite). Such a rescaling may not exist, but it does for instance if the graph has polynomial growth for some rescaling. The infinitesimal collapse in this case is a length space and has a natural measure obtained by push-down of Loeb measure. Does this collapse have any relation to the graph-limit construction?

The two limits are analogous, but correspond to different scaling regimes. For the usual graph limits, one works with dense graphs, in which there are vertices for some large , and the number of edges is comparable to ; in particular, the typical graph distance between two vertices is of the order of (in fact, unless the graph is pretty much k-partite for some bounded k, the graph distance between most pairs of vertices will be 1 or 2). For the asymptotic cone construction, one works with Cayley graphs in which the number of edges is instead comparable to , and the graph distance between two vertices would be polynomial in (if one is in the polynomial growth case). One then normalises the graph distance to be and takes a limit.

For bounded degree graphs, there is another notion of graph limit, based on what happens with local neighbourhoods of randomly chosen vertices at graph distance O(1). For Cayley graphs of finitely generated groups, this comes down to taking pointwise limits of the set of relations between generators, for instance with generators converges in this sense to with the generators . This would also correspond to the asymptotic cone construction if one did not rescale the metric beforehand.

This is a nice way of looking at these things. Has such a soft proof of Roth’s theorem (reduction to Lebesgue differentiation theorem) appeared before? I’ve always wanted to see such a proof.

In the proof of prop 4, by restricting to the symmetric case you get away with just the trivial case of Brunn-Minkowski in , but if you wanted a similar result for or even I guess you would actually need to say “Brunn-Minkowski” at some point. In general I tend to think of prop 4 as an incarnation of Brunn-Minkowski. (By the way, in the statement of prop 4 I think you want , not .)

Regarding Roth’s theorem, certainly the ergodic theory approach to Roth’s theorem can be routed through the Lebesgue differentiation theorem after reducing to the Kronecker factor there, which is also a compact group, and computing the second Host-Kra measure mu^{[2]} on that group. And in the graph theory approach there is an analogous proof (I think in Lovasz’s book, though I don’t have it handy right now) of Roth’s theorem via Lebesgue differentiation on the unit interval [0,1].

By the way, is it really true in Theorem 1 that is always surjective? In other words, in the proof, is it really true that the image of is a compact subgroup of ? I do agree that the image of is always dense, and this seems to be enough for you (i.e., this still implies that the push-forward measure is translation-invariant). If happens to be metrizable then is surjective by countable saturation.

Oh, thanks for pointing out that subtlety! I am so used to dealing with metrisable situations that I forgot that on non-metrisable spaces countable saturation is sometimes insufficient to guarantee closed images. But as you say, dense image is sufficient for the current application.

Correct me if I’m wrong but in fact I think is *injective*. If we consider just the ultrafinite genuine group case for the moment, then there are lots of measurable characters on the ultraproduct coming from the ultraproduct of the dual groups: enough to separate points, by finite Fourier analysis. This implies that is injective.

Somewhat related to this, I’m not sure I believe that is (Loeb, Borel)-measurable. Though indeed the inverse images of the cylinder sets are measurable, unfortunately these do not generate the Borel sigma-algebra. I don’t have an example to prove this point though, so maybe I’ve just overlooked something. However I do beleive that is (Loeb,Baire)-measurable, since by Stone-Weierstrass every continuous function on is the limit of a sequence of character sums.

Yes, you’re right; I had forgotten about the distinctions between Baire and Borel in the inseparable case; I’ve updated the post to correct this. Regarding injectivity, I think you are right for the “universal” group T constructed in the blog post, but in practice one usually quotients down to a more manageable T (in particular, to a separable T, in order to avoid all these technical issues coming from inseparability), at which point injectivity is lost (but surjectivity is gained!). (But conclusion (iv) of the theorem need not hold for all functions f,g after such a quotienting; one can only save this conclusion for a separable space of functions, though this is usually plenty for applications.)

In this paper I give a generalisation of the graph limit setting to the additive setting. I discuss how to take limits of subsets (or more generally functions) in compact abelian groups. Quite surprisingly the limit objects are defined on nil manifolds that are non-commutative structures.

Another very recent paper (joint with Lovasz) is about the non commutative setting:

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